THE GEOMETRIC CAUCHY PROBLEM FOR THE MEMBRANE

arXiv:1406.5981v1 [math.DG] 23 Jun 2014

THE GEOMETRIC CAUCHY PROBLEM FOR THE MEMBRANE SHAPE EQUATION GARY R. JENSEN, EMILIO MUSSO, AND LORENZO NICOLODI Abstract. We address the geometric Cauchy problem for surfaces associated to the membrane shape equation describing equilibrium configurations of vesicles formed by lipid bilayers. This is the Euler–Lagrange equation of the Canham–Helfrich–Evans elastic curvature energy subject to constraints on the enclosed volume and the surface area. Our approach uses the method of moving frames and techniques from the theory of exterior differential systems.

1. Introduction Lipid bilayers are the basic elements of biological membranes and constitute the main separating structure of living cells. In aqueous solution, lipid bilayers typically form closed surfaces or vesicles (closed bilayer films) which exhibit a large variety of different shapes, such as the non-spherical, biconcave shape characteristic of red blood cells [11, 19, 31]. Since in most biologically significant cases the width of the membrane exceeds the thickness by several orders of magnitude, the membrane can be regarded as a two-dimensional surface S embedded in three-dimensional space R3 , with enclosed volume V (S) and surface area A(S). A continuum-mechanical description of lipid bilayers which has been shown to be predictive with respect to the observed shapes dates from the work of Canham [9], Helfrich [16], and Evans [12] at the beginning of the 1970s. In the Canham– Helfrich–Evans model, an equilibrium configuration S, at fixed volume and surface area, is determined by minimization of the elastic bending energy [16, 17, 26, 29, 30, 32, 33, 35, 36] Z Z k 2 ¯ (2H − c0 ) dA + k KdA + pV (S) + λA(S). (1.1) F (S) = 2 S S Here H = (a + c)/2 and K = ac are the mean and Gauss curvatures of the surface S, where a and c denote the principal curvatures, dA is the area element, and k, ¯ c0 , p, and λ are constants: k, k¯ are material dependent elasticity parameters, c0 k, is the spontaneous curvature, p denotes the difference between the outside and the inside pressure, and λ is the surface lateral tension. The pressure p and the tension λ play the role of Lagrange multipliers for the constraints of constant volume and surface area.

2000 Mathematics Subject Classification. 53C42, 58A17, 76A20. Key words and phrases. Biomembranes, Cauchy problem, Helfrich functional, exterior differential systems, lipid bilayers, elasticity, spontaneous curvature, parallel frames, involution, Willmore surfaces, membrane shape equation, Bj¨ orling problem. Authors partially supported by PRIN 2010-2011 “Variet` a reali e complesse: geometria, topologia e analisi armonica”; FIRB 2008 “Geometria Differenziale Complessa e Dinamica Olomorfa”; and by the GNSAGA of INDAM. 1

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GARY R. JENSEN, EMILIO MUSSO, AND LORENZO NICOLODI

The Euler–Lagrange equation of the functional (1.1), often referred to as the membrane shape equation or the Ou-Yang–Helfrich equation, takes the form [27, 28] (1.2) 2k ∆H + 2H(H 2 − K) − 2λ − kc20 H + 2kc0 K − p = 0,

where ∆ denotes the Laplace–Beltrami operator of the induced metric on S. This implies that an equilibrium shape surface S satisfies a fourth-order nonlinear partial differential equation of the form (1.3)

∆H = Φ(a, c),

where Φ is a real analytic symmetric function of the principal curvatures a, c. Examples of surfaces associated to (1.2) include Willmore surfaces [4, 8, 34, 37], which are solutions of the differential equation (1.4)

∆H + 2H(H 2 − K) = 0.

Willmore surfaces are best known in connection to the celebrated Willmore conjecture, recently confirmed by Marques and Neves [20]. The aim of the present paper is to solve the geometric Cauchy problem for the class of surfaces in R3 satisfying the membrane shape equation. More precisely, we shall prove the following. Theorem 1. Let α : J = (a, b) → R3 be a real analytic curve with kα′ (x)k = 1. Let (T = α′ , N , B) be the Frenet frame field along α, with Frenet–Serret equations T ′ = κN,

N ′ = −κT + τ B,

B ′ = −τ N,

where κ(x) 6= 0 is the curvature and τ (x) is the torsion of α. For x0 ∈ J, consider the unit normal vector W0 = N (x0 ) cos a0 + B(x0 ) sin a0 . Let h, hW : J → R be two real analytic functions and assume that Z x τ (u)du + a0 < 0. h + κ sin − x0

Then, there exists a real analytic immersion f : Σ → R3 , where Σ ⊂ R2 is an open neighborhood of J × {0}, with principal curvature line coordinates (x, y), such that: (1) the mean curvature H of f satisfies ∆H = Φ(a, c); (2) (3) (4) (5)

the restriction f |J = α; the tangent plane to f at f (x0 , 0) is spanned by T (x0 ) and W0 ; α is a curvature line of f ; W H|J = h and ∂H ∂y J = h .

ˆ → R3 is any other principal immersion satisfying the above Moreover, if fˆ : Σ ˆ = fˆ(Σ ∩ Σ). ˆ conditions, then f (Σ ∩ Σ) The Cauchy problem addressed in Theorem 1 can be viewed as a generalization of similar problems for constant mean curvature surfaces in R3 and for constant mean curvature one surfaces in hyperbolic 3-space [5, 13], which in turn are both inspired by the classical Bj¨orling problem for minimal surfaces in R3 [25]. Recently, the geometric Cauchy problem has been investigated for several surface classes and in different geometric situations (see, for instance, [1, 2, 6] and the references therein). However, our approach to the problem is different in that we use techniques from the Cartan–K¨ ahler theory of Pfaffian differential systems and the method of

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moving frames (see [21] and [22, 23, 24] for a similar approach to the integrable system of Lie-minimal surfaces and other systems in submanifold geometry). This work was mainly motivated by a paper of Tu and Ou-Yang [35], in which the authors propose a geometric scheme to discuss the questions of the shape and stabilities of biomembranes within the framework of exterior differential forms. The first step in our discussion consists in the construction of a Pfaffian differential system (PDS) whose integral manifolds are canonical lifts of principal frames along surfaces satisfying the membrane shape equation (1.3). We then compute the algebraic generators of degree two for such a PDS and show that the polar space of a 1-dimensional integral element is 2-dimensional. Next, by the Cartan–K¨ ahler theorem we deduce the existence of a unique real analytic integral surface passing through a real analytic integral curve. Finally, we build 1-dimensional integral curves from arbitrary real analytic space curves and two real analytic functions. The proof of Theorem 1 follows from a suitable geometric interpretation of these results. Interestingly enough, in the proof of Theorem 1, a crucial role is played by the use of a relatively parallel adapted frame of Bishop [3] along the given curve α. The paper is organized as follows. Section 2 introduces some background material and recalls the basic facts about Pfaffian differential systems in two independent variables. Section 3 constructs the PDS for the class of surfaces associated to the membrane shape equation. Section 4 proves that this PDS is in involution. Finally, Section 5 proves Theorem 1. For the subject of exterior differential systems we refer the reader to the monographs [7, 10, 14, 15, 18]. The summation convention over repeated indices is used throughout the paper.

2. Background material 2.1. The Euclidean group and the structure equations. Let E(3) = R3 ⋊ SO(3) be the Euclidean group of proper rigid motions of R3 . A group element of E(3) is an ordered pair (P, A), where P ∈ R3 and A is a 3 × 3 orthogonal matrix with determinant one. If we let Aj ∈ R3 , j = 1, 2, 3, denote the j-th column vector of A and regard P and the Aj as R3 valued functions on E(3), there exist unique left invariant 1-forms θi and θji , i, j = 1, 2, 3, such that (2.1)

dP = θi Ai ,

dAj = θji Ai ,

j = 1, 2, 3.

The 1-forms θi , θji are the Maurer–Cartan forms of E(3). Differentiating the orthogonality condition Ai · Aj = δij yields θji = −θij , i, j = 1, 2, 3. These are the only relations among the Maurer–Cartan forms and then (θ1 , θ2 , θ3 , θ12 , θ13 , θ23 ) is a basis for the space of left invariant 1-forms on E(3). Differentiating (2.1), we obtain the structure equations of E(3)

(2.2)

 1 2 2 3 3  dθ = θ1 ∧ θ + θ1 ∧ θ , 2 2 1 3 dθ = −θ1 ∧ θ + θ2 ∧ θ3 ,   3 dθ = −θ13 ∧ θ1 − θ23 ∧ θ2 ,

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and  3 3 2  dθ1 = θ2 ∧ θ1 , dθ13 = −θ23 ∧ θ12 ,   3 dθ2 = θ13 ∧ θ12 .

(2.3)

2.2. Principal frames and invariants. Let f : X → R3 be a smooth immersion of a connected, orientable 2-dimensional manifold X, with unit normal vector field n. Consider the orientation of X induced by n from the orientation of R3 . Suppose that f is free of umbilic points and with the given normal vector such that the principal curvatures a and c satisfy a > c. A principal frame field along f is a map (f, A) : U → E(3) defined on some open connected set U ⊂ X, such that, for each ζ ∈ U , the tangent space df (Tζ X) = span{A1 (ζ), A2 (ζ)}, A3 (ζ) = n(ζ), and A1 (ζ), A2 (ζ) are along the principal directions corresponding to a(ζ) and c(ζ), respectively. Any other principal frame field on U is of the form (f, (±A1 , ±A2 , A3 )). Thus, if X is simply connected, possibly passing to a double cover, we may assume the existence of a globally defined principal frame field (f, A) along f . Following the usual practice in the method of moving frames we use the same notation to denote the forms on E(3) and their pullbacks via (f, A) on X. Let (f, A) be a globally defined principal frame field along f . Then on X, (θ1 , θ2 ) defines a coframe field, θ3 vanishes identically, and θ13 = aθ1 ,

θ23 = cθ2 ,

θ12 = pθ1 + qθ2 ,

where a > c are the principal curvatures and p, q are smooth functions, the Christoffel symbols of f with respect to (θ1 , θ2 ). The structure equations of E(3) give dθ1 = pθ1 ∧ θ2 ,

(2.4)

dθ2 = qθ1 ∧ θ2 ,

the Gauss equation (2.5)

dp ∧ θ1 + dq ∧ θ2 + (ac + p2 + q 2 )θ1 ∧ θ2 = 0,

and the Codazzi equations ( da ∧ θ1 + p(c − a)θ2 ∧ θ1 = 0, (2.6) dc ∧ θ2 + q(c − a)θ1 ∧ θ2 = 0. With respect to the principal frame field (f, A), the first and second fundamental forms of f are given by I = df · df = θ1 θ1 + θ2 θ2 ,

II = −df · dn = aθ1 θ1 + cθ2 θ2 .

For any smooth function g : X → R, we set dg = g1 θ1 + g2 θ2 ,

(2.7)

where the functions g1 , g2 : X → R play the role of partial derivatives relative to θ1 , θ2 . In general, mixed partials are not equal, but satisfy (2.8)

g12 − g21 = pg1 + qg2 ,

by (2.4). Using the relation (∆g) θ1 ∧ θ2 = d ∗ dg = d(−g2 θ1 + g1 θ2 ), which defines the Laplace–Beltrami operator ∆ of the metric I on X, we find ∆g = g11 + g22 + qg1 − pg2 .

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In this formalism, the Gauss and Codazzi equations can be written as  2 2  p2 − q1 = ac + p + q , (2.9) a2 = −p(c − a),   c1 = −q(c − a).

From this we see that there exists a smooth function r, such that ( p2 = r + 21 (ac + p2 + q 2 ), (2.10) q1 = r − 12 (ac + p2 + q 2 ).

Differentiating the structure equations (2.9) and using (2.7) and (2.8), we get  a21 = (p1 − pq)(a − c) + pa1 , a12 = 2pa1 + p1 (a − c),     a22 = r + 21 (ac + p2 + q 2 ) (a − c) + p2 (a − c) − pc2 , (2.11)  c21 = q2 (a − c) − 2qc2 , c12 = (q2 + pq)(a − c) − qc2 ,    c11 = r − 21 (ac + p2 + q 2 ) (a − c) + qa1 − q 2 (a − c).

Applying these formulae, we are now able to express the Laplace–Beltrami operator of the mean curvature of the immersion f in terms of the functions p, q, a, c, r, a1 , c2 , 1 (a11 + c22 ) − r(c − a) + qa1 − pc2 . 2 Thus, f satisfies the differential relation ∆H = Φ(a, c) if and only if

(2.12)

∆H =

(2.13)

a11 + c22 = 2 (Ψ(p, q, a, c, a1 , c2 ) + r(c − a))

where Ψ(p, q, a, c, a1 , c2 ) = Φ(a, c) + pc2 − qa1 . 2.3. Pfaffian differential systems in two independent variables. In this section we recall some basic facts about the Cartan–K¨ ahler theory of Pfaffian differential systems in two independent variables. Let M be a smooth manifold and let ω1 , ω2 , η1 , . . . , ηk , π1 , . . . , πs

be a coframe field on M . Consider the 2-form Ω = ω 1 ∧ ω 2 , which is referred to as the independence condition, and let I = {η 1 , . . . , η k , dη 1 , . . . , dη k } be the ideal of the algebra of exterior differential forms on M generated by the 1-forms η 1 , . . . , η k and the 2-forms dη 1 , . . . , dη k . The pair (I, Ω) is called a Pfaffian differential system in two independent variables. A 2-dimensional integral manifold (integral surface) of (I, Ω) is an immersed 2-dimensional manifold X ⊂ M , not necessarily embedded, such that Ω|X 6= 0,

η j |X = dη j |X = 0,

j = 1, . . . , k.

Similarly, a 1-dimensional integral manifold (integral curve) of the Pfaffian system (I, Ω) is an immersed 1-dimensional manifold A ⊂ M , such that ω 1 ω 1 + ω 2 ω 2 |A 6= 0,

η j |A = 0,

j = 1, . . . , k.

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A 1-dimensional integral element of (I, Ω), at a fixed point m ∈ M , is a 1dimensional subspace [ξ] of Tm M , spanned by a non-zero tangent vector ξ, such that ω 1 (ξ)ω 1 (ξ) + ω 2 (ξ)ω 2 (ξ) 6= 0, η j (ξ) = 0, j = 1, . . . , k. Similarly, a 2-dimensional integral element of (I, Ω) is a 2-dimensional subspace W of Tm M , such that Ω|W 6= 0,

η j |W = dη j |W = 0,

j = 1, . . . , k.

Given a 1-dimensional integral element [ξ], its polar space H([ξ]) is the subspace of Tm M defined by the linear equations: η j = 0,

iξ dη j = 0,

j = 1, . . . , k.

If (2.14)

dim H([ξ]) = 2,

and Ω|H([ξ]) 6= 0,

then H([ξ]) is the unique 2-dimensional integral element containing [ξ]. The following statement is a consequence of the general Cartan–K¨ ahler theorem for exterior differential systems in involution [7, 10]. Theorem 2. Let M be a real analytic manifold and assume the the coframe field ω1 , ω2 , η1 , . . . , ηk , π1 , . . . , πs

is real analytic. If condition (2.14) is satisfied for every 1-dimensional integral element, then for any real analytic 1-dimensional integral curve A ⊂ M there exists a 2-dimensional integral manifold X ⊂ M such that A ⊂ X. The manifold X is ˜ is another 2-dimensional integral manifold passing unique, in the sense that if X ˜ ∩U = through A, then there exists an open neighborhood U ⊂ M of A such that X X ∩ U. 3. The Pfaffian differential system of lipid bilayer membranes 3.1. Principal frames. Let Y(1) be the 10-dimensional manifold Y(1) = E(3) × (p, q, a, c) ∈ R4 | a − c > 0 . Let

θ1 , θ2 , θ3 , θ12 , θ13 , θ23 be the basis of left invariant Maurer–Cartan forms of E(3) which satisfy structure equations (2.2) and (2.3). On Y(1) consider the Pfaffian differential system (I1 , Ω) differentially generated by the 1-forms (3.1)

α1 = θ3 , α3 = θ13 − aθ1 ,

α2 = θ12 − pθ1 − qθ2 , α4 = θ23 − cθ2 ,

with independence condition Ω = θ1 ∧ θ2 . Remark 1. The integral manifolds of (I1 , Ω) are the second order prolongations of umbilic free immersed surfaces in R3 , i.e., smooth maps F(1) := (F, A, p, q, a, c) : X → Y(1) defined on an oriented, connected 2-dimensional manifold X such that: • F : X → R3 is an umbilic free smooth immersion; • (F, A) = (F, (A1 , A2 , A3 )) : X → E(3) is a principal frame field along F ;

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• (θ1 , θ2 ) is a positively oriented orthonormal coframe, dual to the trivialization of dF (T (X)) defined by the tangent vector fields A1 , A2 along F ; • θ1 θ1 + θ2 θ2 is the first fundamental form induced by F ; • A1 , A2 are tangent to the principal curvature lines of F ; • θ12 = pθ1 + qθ2 is the Levi Civita connection with respect to (θ1 , θ2 ); • a and c are the principal curvatures of F and a > c; • aθ1 θ1 + cθ2 θ2 is the second fundamental form of F ; • A3 is the Gauss map of F . Note that F(1) is uniquely determined by the orientation of X, by the umbilic free immersion F , and by the assumption that a > c. From the structure equations (2.2) and (2.3), we obtain, modulo the algebraic ideal {αj } generated by the 1-forms α1 , . . . , α4 , ( dθ1 ≡ pθ1 ∧ θ2 , (3.2) mod {αj } dθ2 ≡ qθ1 ∧ θ2 , and

(3.3)

 1 dα    dα2  dα3    4 dα

≡ 0, ≡ −dp ∧ θ1 − dq ∧ θ2 − (ac + p2 + q 2 )θ1 ∧ θ2 , ≡ −da ∧ θ1 + p(c − a)θ1 ∧ θ2 , ≡ −dc ∧ θ2 − q(c − a)θ1 ∧ θ2 .

mod {αj }

3.2. Prolongation. Let Y(2) be the 17-dimensional manifold Y(2) = Y(1) × (p1 , q2 , r, a1 , c2 , a11 , c22 ) ∈ R7 , where

(p1 , q2 , r, a1 , c2 , a11 , c22 ) are the new fiber coordinates. Consider on Y(2) the Pfaffian differential system (I2 , Ω) differentially generated by the 1-forms (α1 , . . . , α4 , β 1 , β 2 , γ 1 , γ 2 , δ 1 , δ 2 ), where α1 , . . . , α4 are defined as in (3.1) and  1 2 2 2 1 1  β = dp − p1 θ − (r + 2 (ac + p + q ))θ ,   1  β 2 = dq − (r − 2 (ac + p2 + q 2 ))θ1 − q2 θ2 ,   γ 1 = da − a θ1 + p(c − a)θ2 , 1 (3.4) 2  γ = dc + q(c − a)θ1 − c2 θ2 ,      δ 1 = da1 − a11 θ1 + (−2a1 p + (c − a)p1 )θ2 ,    2 δ = dc2 + (2c2 q + (c − a)q2 )θ1 − c22 θ2 .

From (3.3) and (3.4) it follows that, modulo the algebraic ideal generated by αj , β a , γ a and δ a , j = 1, 2, 3, 4; a = 1, 2, we have ( dαj ≡ 0, j = 1, 2, 3, 4, (3.5) mod {αj , β a , γ a , δ a } dγ a ≡ 0, a = 1, 2,

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Remark 2. The integral manifolds of (I2 , Ω) are fourth order prolongations of immersed surfaces in R3 , that is, smooth maps F(2) := (F, A, p, q, a, c, p1 , q2 , r, a1 , c2 , a11 , c22 ) : X → Y(2) , where (F, A, p, q, a, c) : X → Y(1) is an extended principal frame along F and the functions p1 , q2 , r, a1 , c2 , a11 , c22 : X → R are defined by  dp = p1 θ1 + p2 θ2 , dq = q1 θ1 + q2 θ2 ,    r = p − 1 (ac + p2 + q 2 ) = q + 1 (ac + p2 + q 2 ), 1 2 2 2 1 2 1  da = a θ + a θ , dc = c θ + c2 θ 2 , 1 2 1    da1 = a11 θ1 + a12 θ2 , dc2 = c21 θ1 + c22 θ2 .

Note that F(2) is uniquely determined by the orientation of X, by the immersion F , and by the assumption that the principal curvature satisfy a > c. The canonical prolongations of smooth immersions F : X → R3 satisfying the partial differential relation ∆H = Φ(a, c) are characterized by the equation (3.6)

a11 + c22 = 2 [Φ(a, c) + r(c − a) − qa1 + pc2 ] .

3.3. The Pfaffian differential system of lipid bilayer membranes. Let Y∗ be the 16-dimensional real analytic submanifold of Y(2) defined by a11 + c22 = 2 [Φ(a, c) + r(c − a) − qa1 + pc2 ] . On Y∗ we consider the fiber coordinates p, q, a, c, p1 , q2 , r, a1 , c2 , ℓ, where ℓ is given by (3.7)

(

a11 = ℓ + r(c − a) + Ψ(p, q, a, c, a1 , c2 ), c22 = −ℓ + r(c − a) + Ψ(p, q, a, c, a1 , c2 ),

where Ψ(p, q, a, c, a1 , c2 ) = Φ(a, c) − qa1 + pc2 . Definition 1. The restriction of (I2 , Ω) to Y∗ is denoted by (I∗ , Ω) and is referred to as the Pfaffian exterior differential system of lipid bilayer membranes satisfying the differential relation ∆H = Φ(a, c). Remark 3. The integral manifolds of (I∗ , Ω) are canonical prolongations of umbilic free immersions F : X → R3 whose mean curvature H satisfies ∆H = Φ(a, c).

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4. Involution 4.1. Algebraic generators. By construction, the Pfaffian differential system I∗ is generated, as a differential ideal, by the restrictions to Y∗ of the 1-forms αj , β a , γ a , δ a , j = 1, . . . , 4; a = 1, 2. The generators αj , β a , γ a are expressed in terms of the fiber coordinates p, q, a, c, r, a1 , c2 , p1 , q2 as in (3.1) and (3.4), while the generators δ 1 and δ 2 are given by ( δ 1 = da1 − (ℓ + r(c − a) + Ψ) θ1 + (p1 (c − a) − 2a1 p) θ2 , (4.1) δ 2 = dc2 + (q2 (c − a) + 2c2 q) θ1 + (ℓ − r(c − a) − Ψ) θ2 . Observe that (θ1 , θ2 , α1 , . . . α4 , β 1 , β 2 , γ 1 , γ 2 , δ 1 , δ 2 , dp1 , dq2 , dr, dℓ) is a global coframe field on Y∗ . Let ∂θ1 , ∂θ2 , ∂α1 , . . . , ∂α4 , ∂β 1 , ∂β 2 , ∂γ 1 , ∂γ 2 , ∂δ1 , ∂δ2 , ∂dp1 , ∂dq2 , ∂dr , ∂dℓ denote its dual basis on T (Y∗ ). According to (3.5), we have ( dαj ≡ 0, j = 1, 2, 3, 4, (4.2) mod {αj , β a , γ a , δ a }. dγ a ≡ 0, a = 1, 2,

Thus, the differential ideal I∗ is algebraically generated by αj , β a , γ a , δ a , j = 1, 2, 3, 4; a = 1, 2, and by the exterior differential 2-forms dβ a , dδ a , a = 1, 2. Using (3.2), (3.4), (4.1) and (4.2), we obtain, modulo the algebraic ideal {αj , β a , γ a , δ a },  1 dβ ≡ −dp1 ∧ θ1 − dr ∧ θ2 − B 1 θ1 ∧ θ2 ,    dβ 2 ≡ −dr ∧ θ1 − dq ∧ θ2 − B 2 θ1 ∧ θ2 , 2 (4.3) 1 1  dδ ≡ −dℓ ∧ θ − (c − a)dr ∧ θ1 + (c − a)dp1 ∧ θ2 − D1 θ1 ∧ θ2 ,    2 dδ ≡ (c − a)dq2 ∧ θ1 + dℓ ∧ θ2 − (c − a)dr ∧ θ2 + D2 θ1 ∧ θ2 , where B a and Da , a = 1, 2, are real analytic functions of the fiber coordinates.

Remark 4. Note that if Ψ is polynomial, the B a and the Da are polynomial functions of the fiber coordinates. This is still true of the B a , but not of the Da , in the event that Ψ is not a polynomial. In conclusion, we have proved that (I∗ , Ω) is algebraically generated by αj , β a , γ , δ a and by a

(4.4)

 1 Ω     Ω2  Ω3    4 Ω

= dp1 ∧ θ1 + dr ∧ θ2 + B 1 θ1 ∧ θ2 , = dr ∧ θ1 + dq2 ∧ θ2 + B 2 θ1 ∧ θ2 , = dℓ ∧ θ1 + (c − a)dr ∧ θ1 − (c − a)dp1 ∧ θ2 + D1 θ1 ∧ θ2 , = (c − a)dq2 ∧ θ1 + dℓ ∧ θ2 − (c − a)dr ∧ θ2 + D2 θ1 ∧ θ2 .

4.2. Polar equations. Let [ξ] be a 1-dimensional integral element, where ξ ∈ Tm Y∗ is a tangent vector of the form ξ = x1 ∂θ1 + x2 ∂θ2 + x3 ∂dp1 + x4 ∂dq2 + x5 ∂dr + x6 ∂dℓ ,

(x1 )2 + (x2 )2 6= 0.

Its polar space H([ξ]) is the subspace tangent to Y∗ defined by the polar equations αj = 0,

β a = 0,

γ a = 0,

δ a = 0,

iξ Ωj = 0,

j = 1, . . . , 4; a = 1, 2.

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From (4.4), it follows that the polar equations are linearly independent provided c−a 6= 0, for every 1-dimensional integral element [ξ]. This implies that H([ξ]) is the unique 2-dimensional integral element containing [ξ]. Thus, by the Cartan–K¨ ahler theorem we have the following. Theorem 3. The Pfaffian differential system (I∗ , Ω) is in involution and its general solutions depend on four functions in one variable. More precisely, for every, 1dimensional real analytic integral manifold A ⊂ Y∗ there exists a real analytic 2-dimensional integral manifold X ⊂ Y∗ passing through A. In addition, the 1dimensional integral manifolds depend on four functions in one variable. Remark 5. The 2-dimensional integral manifold X passing through A is unique, in the sense that if X˜ ⊂ Y∗ is another real analytic 2-dimensional integral manifold containing A, then X and X˜ agree in a neighborhood of A. 5. The proof of Theorem 1 We are now ready to prove Theorem 1. We are given the Cauchy data α, x0 , W0 , h, hW , consisting of a real analytic curve α : J → R3 , a point x0 ∈ J, a unit normal vector W0 = cos a0 N (x0 ) + sin a0 B(x0 ), and two real analytic functions h, hW : J → R3 as in the statement of Theorem 1. Set Z x τ (u)du + a0 , s(x) = − x0

so that s : J → R is real analytic and s(x0 ) = a0 , and define m := −h − κ sin s(x) > 0. Let W : J → R3 be the unit normal vector field along α defined by W = cos s(x)N + sin s(x)B. Consider the positively oriented orthonormal frame field given by (5.1)

G = (α, T, W, JW ) : J → E(3),

where JW = − sin s(x)N + cos s(x)B. Using the Frenet–Serret equations T ′ = κN , N ′ Frenet frame (T, N, B), it is easily seen that  0 0 0 1 0 −p dG =G 0 p 0 dx 0 a 0

= −kT + τ B, B ′ = −τ N for the  0 −a , 0 0

where p, a : J → R are the real analytic functions p = κ cos s(x),

a = −κ sin s(x).

Remark 6. The frame field (T, W, JW ) is a relatively parallel adapted frame along the curve α in the sense of Bishop [3].

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Next, if we set   c = a − 2m,    1 dc  q = − c−a   dx ,   da  a1 = dx ,    c = 2hW + p(c − a), 2 dp  p 1 = dx ,    dc2 1  q2 = − c−a  dx + 2c2 q ,   dq  + 21 (ac + p2 + q2 ), r = dx    2  d a l = dx 2 − r(c − a) − Ψ(p, q, a1 , c2 ),

(5.2)

then (5.3)

A : J → Y∗ , x 7→ (G, p, q, a, c, p1 , q2 , r, a1 , c2 , l) |x

is a 1-dimensional integral manifold of I∗ such that θ1 = dx,

θ2 = 0,

defined by   p ◦ A = p, q ◦ A = q,    a ◦ A = a, c ◦ A = c,    p ◦ A = p , q ◦ A = q , 1 1 2 2  r ◦ A = r,      a1 ◦ A = a1 , c2 ◦ A = c2 ,    l ◦ A = l.

(5.4)

Definition 2. We call U := Im A ⊂ Y∗ the canonical 1-dimensional integral manifold defined by the Cauchy data α, x0 , W0 , h, hW . For a set of Cauchy data α, x0 , W0 , h, hW , Theorem 3 and Remark 5 imply that there exists a unique real analytic integral manifold X ⊂ Y∗ of (I ∗ , Ω) such that U ⊂ X , where U is the integral curve defined by α, x0 , W0 , h, hW . On X we consider the orientation defined by the 2-form Ω = θ1 ∧ θ2 . The map F : X ∋ (P, A, p, q, a, c, p1 , q2 , r, a1 , c2 , ℓ) 7→ P ∈ R3 is a real analytic immersion and, by construction, its prolongation F(2) coincides with the inclusion map ι : X → Y∗ . According to Remarks 1 and 2, we infer that • X ∋ (P, A, p, q, a, c, p1 , q2 , r, a1 , c2 , ℓ) 7→ (P, A) ∈ E(3) is a principal frame field along F ; • F satisfies ∆H = Φ(a, c); • (θ1 , θ2 ) is a positively oriented orthonormal principal coframe on X along the immersion F ; • a, c : X → R are the principal curvatures of F and dH =

1 1 1 (da + dc) = (a1 − q(c − a))θ1 + (c2 − p(c − a))θ2 . 2 2 2

Since X satisfies the initial condition U ⊂ X and A∗ (θ2 ) = 0, we can state the following.

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Lemma 4. A : J → X is a curvature line of F such that F ◦ A = α,

A ◦ A = G.

In particular, we have F∗ (TA(x0 ) (X ) = span (A1 (α(x0 )), A2 (α(x0 )) = span (T (x0 ), W (x0 )). From (3.4), (5.3) and (5.4), we obtain (5.5)

H ◦A=

1 (a + c) = h 2

and 1 (c2 − p(c − a))θ2 |A(x0 ) = hW θ2 |A(x0 ) , mod θ1 |A(x0 ) . 2 Let X1 , X2 denote the frame field dual to the coframe field θ1 , θ2 on the integral manifold X and let Θ be the local flow generated by the nowhere vanishing vector field X2 . Then Θ is a real analytic map Θ : U → X defined on an open neighborhood U ⊂ X × R and the set

(5.6)

dH|A(x0 ) ≡

Σ := {(x, y) ∈ J × R | (A(x), y) ∈ U} is an open neighborhood of J × {0} ⊂ R2 . Using the immersion F and the flow Θ, define the map f : Σ → R3 by f (x, y) = F (Θ(A(x), y)) . 3

The map f : Σ → R is a real analytic immersion such that f (x, 0) = F (Θ(A(x), 0)) = α(x), ∀ x ∈ J. By construction, f is a re-parametrization of F and hence satisfies the differential relation (1.3). Lemma 4 and the equations (5.5) and (5.6) imply that f satisfies ˆ → R3 is any other the required conditions (3), (4) and (5) of Theorem 1. If fˆ : Σ ˆ → Y∗ is immersion satisfying the same conditions, then its prolongation fˆ(2) : Σ an integral manifold of (I2 , Ω) passing through U. Then, by the uniqueness part of ˆ = fˆ(2) (Σ ∩ Σ). ˆ This concludes Cartan–K¨ ahler theorem, it follows that f(2) (Σ ∩ Σ) the proof of Theorem 1. References [1] L. J. Al´ıas, R. M. B. Chaves, and P. Mira, Bj¨ orling problem for maximal surfaces in Lorentz-Minkowski space, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 2, 289–316. ´ lvez, The Cauchy problem for improper [2] J. A. Aledo, R. M. B. Chaves, and J. A. Ga affine spheres and the Hessian one equation, Trans. Amer. Math. Soc. 359 (2007), no. 9, 4183–4208. [3] R. L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly 82 (1975), 246–251. [4] W. Blaschke, Vorlesungen u ¨ber Differentialgeometrie und geometrische Grundlagen von Einsteins Relativit¨ atstheorie, B. 3, bearbeitet von G. Thomsen, J. Springer, Berlin, 1929. [5] D. Brander and J. F. Dorfmeister, The Bj¨ orling problem for non-minimal constant mean curvature surfaces, Comm. Anal. Geom. 18 (2010), no. 1, 171–194. [6] D. Brander and M. Svensson, The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps, J. Differential Geom. 93 (2013), no. 1, 37–66. [7] R. L. Bryant, S.-S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior differential systems, MSRI Publications, 18, Springer-Verlag, New York, 1991. [8] R. L. Bryant, A duality theorem for Willmore surfaces, J. Differential Geom. 20 (1984), 23–53.

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(G. R. Jensen) Department of Mathematics, Washington University, One Brookings Drive, St. Louis, MO 63130, USA E-mail address: [email protected] (E. Musso) Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy E-mail address: [email protected] ` degli Studi di (L. Nicolodi) Dipartimento di Matematica e Informatica, Universita Parma, Parco Area delle Scienze 53/A, I-43100 Parma, Italy E-mail address: [email protected]